Abstract
In this article we study the relations between three classes of lattices each extending the class of distributive lattices in a different way. In particular, we consider join-semidistributive, join-extremal and left-modular lattices, respectively. Our main motivation is a recent result by Thomas and Williams proving that every semidistributive, extremal lattice is left modular. We prove the converse of this on a slightly more general level. Our main result asserts that every join-semidistributive, left-modular lattice is join extremal. We also relate these properties to the topological notion of lexicographic shellability.
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Mühle, H. Extremality, left-modularity and semidistributivity. Algebra Univers. 84, 16 (2023). https://doi.org/10.1007/s00012-023-00814-8
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DOI: https://doi.org/10.1007/s00012-023-00814-8